Saturday, August 18, 2007
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I've seen a lot of flack aimed at string theory criticizing its supposed untestability. What's the problem here, in your view? Is it just that conducting experiments in this subfield would require inordinately expensive equipment (like particle physics, only more so), or is there some larger issue?

String theory is a theory of quantum gravity. It's been known for decades that the characteristic effects of quantum gravity only appear in "full strength" in extreme conditions - characterized by the so-called Planck scale. The typical distance where stringy or any quantum gravitational processes are easily visible is the supertiny Planck length 10^{-35} meters, the characteristic temperature is the Planck temperature 10^{32} Kelvin, and so forth.

The only known loophole are the braneworlds i.e. the models of old large dimensions and warped extra dimensions in which the characteristic phenomena of string theory or quantum gravity may be much more accessible, even by the LHC. But these scenarios look unlikely.

If we ignore this possibility, a collider that would directly test quantum gravity would be as large as the visible Universe. Since the very early days of quantum gravity, it was clear that the attack on its puzzles would be theoretical in character and the field has been a domain of some of the "purest" theorists - even though some people seem to be surprised even in 2007. What was not known is that the mathematical consistency criteria almost certainly single out not only a pretty answer but a unique answer - one that is called string theory for historical reasons - even though the repertoire of physical phenomena and solutions to this theory is very rich.

According to everything we know, there is no more serious problem than a problem in practice. However, some questions of a similar character may be unanswerable - for example those related to the precise nature of the de Sitter thermal radiation.

What is the observational evidence that requires a theory like dark matter, and why do other theories come up short in comparison to dark matter?

Newton's theory has been a successful theory of gravity. Einstein's general relativity has superseded Newton's theory but in the context of rotation of galaxies, they give pretty much identical answers. While these theories agree with many phenomena, there seemed to be a discrepancy in rotations of galaxies. The density of matter that was deduced from its gravitational effects - the orbital speed as a function of the radial coordinate - was much greater than the density of matter stored in visible stars. The galaxies rotate much like CDs (where the angular velocity is pretty much constant) instead of the Solar System (a prediction of gravity based on visible stars where the angular velocity should be much higher near the center). That implies that either the theory of gravity is wrong (both of them) or that there are new, invisible sources of gravity: they were called dark matter.

One year ago, NASA offered a more direct evidence for dark matter. They found that during a collision, the (reconstructed) dark matter moves pretty much independently from the visible stars. This observation, indicating their independent existence, falsifies a large number of theories that modify the laws of gravity - all the theories where gravity is still associated with visible stars and is additive and positive - because in these theories, the deduced dark matter should still move together with the stars.

Once we adopt that dark matter is real, it must be composed out of a new kind of particle, at least partially. Neutralinos predicted by supersymmetry, a natural part of string theory, are the most obvious (and popular) candidates. But the structure of dark matter is extremely unconstrained - it can be made out of pretty much anything new as long as you adjust the mass and the interaction strength correctly - so one doesn't expect to answer all remaining questions about physics by using the existence of dark matter only. Dark matter is not such a big clue, after all. The fact that its total mass exceeds the mass of all visible matter five-fold doesn't change it into a big clue either. ;-) It's still some boring, dumb matter sitting around galaxies.

Incidentally, dark energy is 70% of the mass of the Universe but this huge mass is not why dark energy is such an unusual hint. Quite on the contrary: the mystery is why the dark energy is so light!

But, more multiverse? Why, oh why?

Because AB seems to prefer to tell people what they want to hear rather than what is true and important. (Sorry if you don't know the context here.)

Do you think the subject did not receive the attention it so richly deserves?

Typical discussions about the anthropic principle in 2007 are even less sophisticated than they used to be three years ago or so. Among the scientists, the progress was close to zero; among the general public, the progress was negative, especially because of a misinformation campaign by certain professional amateurs. ;-)

General clichés about the landscape are the most favorite topic for people with severely limited intelligence who prefer to repeat and condemn, in a circle of similarly handicapped friends, a few meaningless dogmas that they have memorized instead of thinking about new questions, confronting their hypotheses against known facts, and refining their opinions. The actual celebrated recent work in theoretical physics has almost nothing to do with the discussions of these people, for example with the writings of most journalists and visitors of a certain portion of the blogosphere.

Yes, it does. Particles in physics are way more abstract objects than cows or other animals whose number can usually be agreed upon. Much like other concepts in modern physics, they can only be carefully analyzed and interpreted by the language of advanced mathematics. In quantum field theory, a particle is an excitation of a quantum field. A quantum field in a non-interacting theory is mathematically equivalent to a set of infinitely many harmonic oscillators; for each conceivable value of velocity (and each independent polarization if there are many), there is one oscillator.

In quantum mechanics, the harmonic oscillator can only have an energy (above its ground state) that is a multiple of its frequency. The relevant integer is interpreted as the number of particles. To find the probability that the number of particles equals N, you must decompose your wave function(al) into eigenstates of the harmonic oscillator. Some of the readers have heard about the gaussians multiplied by Hermite polynomials.

Accelerated observers can also interpret quantum fields around them as a set of harmonic oscillators but they must use different frequencies for different directions and/or speeds where the particles can move, relatively to the inertial observers. That changes (squeezes) the shape of the wave function(al)s with well-defined numbers of particles. While the number of particles is still quantized, an accelerated observer will end up with different answers because he is using a different Hamiltonian to determine the energy of the harmonic oscillator. He is using a different one because such a new Hamiltonian must evolve "horizontal" time slices in the past into "tilted" time slices in the future. That's why a little bit of the boost generator must be included in the accelerating guy's Hamiltonian.

The simplest example why they won't agree is the vacuum itself. The vacuum is defined as the state of the lowest possible energy. But because two observers that are accelerating with respect to one another use a different definition of energy, they will also disagree about the identity of the vacuum i.e. the ground state of the physical system.

Classically, this disagreement wouldn't arise because the ground state of the classical harmonic oscillator is simply the point x=0, p=0, for any frequency. But the uncertainty principle in quantum mechanics makes these two conditions incompatible. The ground state is replaced by the gaussian wave function. However, the width of this wave function depends on the frequency (for any width, the uncertainty principle is saturated) and the two observers won't agree about this frequency and the width. That's why they won't agree what it means to have empty space.

Additional subtleties about the counting of particles arise when interactions are taken into account. The number of particles is only integer in the limit where the interactions can effectively be neglected. The more a system interacts, the less meaningful it is to count the number of particles. For example, conformal theories where particles interact strongly at many energy scales make it very difficult to count particles by integers - a basic old observation that was recently named "unparticle physics". Counting of particles also depends on the renormalization scale. If you raise it, you will see for example many more gluons inside a proton. In some sense, saying that there are just three quarks inside a proton is an idealized statement associated with the lowest possible energy scale. The higher resolution you take, the more gluons (and quark-antiquark pairs) you see.

Do we expect an electron in free fall to radiate?

Relatively to comoving objects, it doesn't radiate because the situation is equivalent to inertial electrons in empty space by the principle of equivalence, a principle that still holds, at least semiclassically. Relatively to another, accelerating observer - one that is still describing a small region of space only, such an electron does radiate. The reason was explained above: mutually accelerating observers use different Hamiltonians and don't agree about the number of particles.

How does a co-moving observer see a non-radiating static electric field while a stationary observer sees a radiating charge?

Both observers, if they formulate their questions accurately, must define what they mean by states with a certain number of photons. Because of the difference in the Hamiltonians mentioned above, leading to different bases of the Hilbert space of the harmonic oscillators, they will not agree what is a wave function with N photons. Moreover, the amount of their disagreement changes with time. Consequently, they will not agree how many photons were radiated away.

If we think of the radiation as discrete photons, are they there or not?

Photons may be thought of as discrete objects but the idea that they're universal points whose location will be agreed upon by all observers is a misguided artifact of an oversimplified analogy building on classical physics and 18th century intuition. In reality, where their existence is controlled by quantum physics, a photon is the state obtained by the action of a creation operator upon the vacuum state that has no particles. Different observers won't even agree what is the vacuum state, as argued above. Empty space for one observer is a superposition of various states with different numbers of particles moving in different directions according to another observer. They also define the creation operators with different relative coefficients and the relations between them can be easily calculated by robust mathematical laws.

Or does that question make no sense somehow when formulated precisely?

The question whether particle production exists does make sense when everything is formulated accurately but the arguments leading to the conclusion that particle production doesn't exist and different observers will agree about the answer - even though these arguments sound plausible at the verbal level - are shown to be wrong when one defines and counts particles more carefully using the proper mathematical toolbox. Once again, the question was perfectly fine but the answer was completely wrong.

Dead webs

Hmm, is anybody thinking about relaunching the String Coffee Table or SPS?

Not sure about the coffee table but SPS is fully dead because the old server receiving the postings went out of business and computer administrators who would be fixing this problem didn't exist at Harvard.

For instance, how many flavours there are really at the ends of the type I open string? 32, 10, or 5?

In bosonic string theory, the theory with open strings and with the O(8192) gauge group is special because of a tadpole cancellation: see e.g. this paper by the honorary string theorist Steven Weinberg. If it were U(8192), we could talk about 8192 colors and 8192 anti-colors. Because the group is orthogonal, colors and anti-colors are identified with one another, in a sense, so we have 8192 half-colors. A T-dual theory would literally have 4096 colors: it would have 4096 D24-branes plus their 4096 images behind the "mirror" (the orientifold plane). This is because O(8192) can be broken to U(4096). However, the space-filling D25-branes in normal bosonic string theory always overlap with each other and can't be separated, so the discussion is confusing. Whether we say that the number of colors iis 8192 or 4096 is just a linguistic question about our conventions. What matters are equations: O(8192) is the preferred gauge group.

In superstring theory in 10 dimensions, the special number of colors is way more important than in bosonic theory because all anomalies and problems cancel for the SO(32) theory called type I string theory - the background used by Green and Schwarz to initiate the first superstring revolution. Replace 8192, 4096, D25, and D24 by 32, 16, D9, and D8 in the previous paragraph and it will be true for superstring theory. There are 32 half-Chan-Paton factors in type I string theory.

It is possible (but not necessary) to generate Chan-Paton factors from (dynamically trivial) fermionic fields if the number of colors is a power of two. The number 5 refers to the number of fermions that need to be added to the endpoints of strings and if you're interested about the origin of this particular number, see the slow comments. A function expanded in these 5 anticommuting variables will have 32 components that can be conveniently identified with the half-colors. But because the 5 fermionic variables are so dummy, the precise arrangement of the 32 colors into the corners of a five-dimensional cube is mostly an unphysical artifact of this trick.

The powers of two are rather natural for the number of colors in many backgrounds of string theory, an insight that has also been hijacked by computer scientists who have revealed their true colors by not citing string theory. ;-)

Why ghosts are needed for quantization of QCD?

Dear experimentalist, ghost are not needed for "all" quantizations of QCD. Ghosts are just a wise technological trick that allows us to quantize a theory with gauge symmetries covariantly. QCD contains unphysical degrees of freedom because of the gauge invariance. We can eliminate them from the scratch and then ghosts are not needed. Alternatively, we may keep them, to make the theory prettier. Then the path integral - the simplest way to deal with the system - must be divided by the volume of the gauge group. A choice of gauge conditions must be accompanied by a Jacobian and this Jacobian may be expressed as a path integral over a Faddeev-Popov ghost system, as originally found by Feynman. See also an article about BRST invariance.

What is QCD?

QCD is the boundary CFT dual to a particular background of string theory that is important for our understanding of strong interactions.

Twistors

Firstly, is the following statement of Witten's conjecture correct: The only non-zero amplitudes for spin one particle scattering come from configurations of points which lie on algebraic curves in twistor space.

Yes, it's correct for the tree processes as argued in the original Witten's paper, but it is not the most favorite twistor-based approach that is used by particle physicists these days. A more popular and effective approach, due to Cachazo, Svrček, and Witten, is based on Feynman diagrams whose basic building blocks are MHV vertices - corresponding to lines (simplest algebraic curves) only. These two approaches are equivalent.

What is the evidence behind Witten's conjecture, and what has been done to prove/disprove it? What is the present status?

The evidence was originally based on explicit calculations of examples but later, people have proven it. One set of proofs heavily builds on an analysis of poles of the amplitudes and the way how the amplitudes can be reconstructed from the known correct poles, unitarity, and some derivable recursion formulae. Another proof of the MHV version of the twistor rules directly constructs a field redefinition in the light cone gauge so that the gauge theory in the new variables directly leads to the desired Feynman rules.

Does twistor-string theory extend the Atiyah-Hitchin-Singer theorem about instantons (using generalizations of the Ward theorem) to other cases (quantum corrections)?

I am not sure what the extension of the Atiyah-Hitchin-Singer theorem is supposed to be in this case. But the starting point of Witten's expansion is not given by the equations being classical but rather by the gauge fields being self-dual. His approach is a generalization of a self-dual gauge field to a more general gauge field (or superfield).

It's not a question but be careful what you eat. There is no well-defined boundary between "differential geometry relevant for gauge theory" and "independent differential geometry". I would argue that differential geometers should naturally be interested in all results that emerge in gauge theories, especially classical gauge theory, which is why they might be interested in this finding, too. But it seems hard to formulate what's nontrivial about Witten's rules while avoiding gauge theory with its Lagrangian (or an awkward mathematical description of the same thing). His basic statement is about a geometric interpretation of the interacting gauge theory. Nothing interesting is left is you decide to throw gauge theories away, considering them to be dirty physics. ;-)

I'd like to see a discussion of how (or if?) string theory reconciles the geometric interpretation of gravity with the quantum interpretation.

String theory implies that deformations or wiggles on geometry are equivalent to a condensate of strings in a certain oscillation pattern. The infinities and ambiguities are removed because of a combination of two facts: the string is an extended object (which prevents points from getting to close to each other, a qualitative source of short-distance divergences); the internal dynamics of a string is essentially unique and can be described by a finite theory, too.

An analysis of the interactions between strings in these vibration states shows that they are completely equivalent to interactions of quanta of gravitational waves and general relativity follows. See Why are there gravitons in string theory.

For that matter, I'd love to see a discussion about any other problems that string theory solves, as well as new problems created by the theory.

See Top twelve results of string theory for a list of results. I am not aware of problems "created" by the theory per se. I am only aware of two mutually related problems that are also problems for all other theoretical physicists today and that were expected to have been solved except that they're still not answered. What are they?

We don't know how to locate the right minimum in the configuration space of string theory that matches our Universe. And we don't know how to explain the tiny value of the cosmological constant. No other theory can answer these questions either, of course. The champions of the anthropic principle essentially argue that the existence of life is the only and sufficient answer to both questions and no other answer has to be searched for.

My screen freezes everytime that I have tried to access your blog over the last couple of weeks. Any hints on how to reslove this problem?

Use a feed reader, e.g. this page. A Google account recommended. The design is modest, fast, white, and clean. Comments are not there.

As I understand it, the multiverse is created by nucleating new universes as regions of "true vacuum". Question: what does string theory have to say about this mysterious process?

This mysterious process is a paradigm believed by many to be correct because of effective field theory, the long-distance limit of any theory including string theory. That's why the existence of multiverse itself doesn't depend on string theory too much, except for a complicated configuration space or "landscape" that string theory inserts as input for the field-theoretical arguments to make the set of possible transitions richer. That's why many non-string theorists, including Alex Vilenkin or Alan Guth, take the mechanisms behind the multiverse creation seriously.

String theory reproduces the field-theoretical results at long distances and is expected to reproduce the bubbling processes, too. Whether these processes are useful for learning something new, even in the context of field theory, is a different question. My personal guess is No: considering the whole "family tree" of our Universe before it was small is probably unphysical.

While many believe that the qualitative conclusions about the existence of pocket universes won't be influenced by string theory, some thoughtful people such as Tom Banks argue that string theory radically changes the predictions of field theories concerning the ability to tunnel to different vacua. But he or they (and others) haven't yet found a crisp argument that would convince other thoughtful people that this new stringy behavior is real so people conservatively continue to believe whatever seems to be the best conclusion of the previous level of theories, namely local quantum field theory.

Question: is there anything in string theory that makes it more plausible that a patch of true vacuum would necessarily or naturally evolve into something like our world?

A patch of true vacuum naturally evolves into a large flat space similar to ours, following the standard insights of inflationary cosmology that are incorporated in many vacua of string theory. At this moment, there exists no known explanation that the spectrum of particles and forces - and the constants - should naturally be what they are instead of other allowed choices. The advocates of the anthropic reasoning argue that there will never exist a better explanation for these "coincidences", neither in string theory nor elsewhere, and of course, they might be right or wrong.

If we cannot answer these questions, should we believe that the landscape is physically relevant?

Yes. There are many concepts and insights in physics that are believed - or can be shown - to be physically relevant even if they don't allow physicists to answer all related questions at a given moment that they would like to be answered. For example, Schrödinger's equation correctly describes all atoms but it doesn't allow us to explain why it is exactly carbon dioxide that we call life. QCD was always the right theory of nuclei but it didn't allow people to calculate the spectrum of mesons at the beginning even though good people already knew that it was certainly correct. There are many examples.

The lesson is that if one can imagine a particular argument to support a theory, it doesn't mean that it is the only possible method how the theory can be supported or even proven (or disproven, in the case of wrong theories). And if this argument isn't completed, it doesn't mean that science is stuck. There are many other ways to proceed. Other people may be smarter and luckier and find these better ways, instead of being stuck with a wrong approach.

The existence of a large number of AdS universes that are consistent with the laws of quantum gravity and other qualitative features of our Universe - the existence of gauge theories and matter fields - seems well-established whether or not you believe that string theory describes our Universe. We will probably never "unlearn" this fact and return to a narrow-minded box where the solution was thought to be completely unique. The people who imagine that by abandoning string theory, they may solve any of these problems, are victims of wishful thinking because without string theory, the spectrum of possibilities is much greater still.

To a lesser extent, the comment about AdS spaces is also true for supersymmetry-breaking de Sitter vacua: there is probably a lot of them, too.

This fact implies that the "landscape" is the set of candidate loci where our Universe may be. This landscape is less accurate information than the location we would like to know at the end but it is much more concrete and predictive than the space of possibilities according to local field theories, the swampland, and way more concrete and descriptive (and correct) than the ideas about physics that people had 100 years ago.

On the other hand I cannot understand why MR thinks that the anthropic principle isn't interesting scientifically.

It is perhaps interesting scientifically to some extent but there are not too many solid arguments or theories to talk about right now. That's why a discussion about this general topic among two intelligent people cannot take more than two hours. They say everything they want to say and everything is clear to both of them. All people who are spending much more than two hours with these trivial general comments are mentally challenged. For example, a blog called N.E.W. spends with this trivial topic more than 3 years and its owner, just like 99% of his visitors, are still completely unable to understand the very basic facts, or at least they pretend so. This is not how scientists discuss issues: this is how morons talk to each other.

We contribute lots of boring papers to the arxiv too!

There are many boring papers but what is clear is that a person who is not interested in theoretical physics will find almost all of them boring - or well beyond his abilities, to use a more honest and less arrogant description of his actual situation. There is nothing surprising about it and this tautological observation is certainly not a reason that should influence what physicists think about.

I just wanted to stress that while the philosophical stuff about the multiverse has been said too many times, that does not imply that one can no longer have a meaningful discussion about the science of it.

The implication is not necessarily robust but the conclusion is certainly true: there has been no meaningful scientific discussion about the issue of the anthropic principle in the blogosphere and most media, largely because the people who want to ignite such discussions are [an accurate but potentially offensive noun was removed], and one shouldn't expect this situation to change with the same amount of knowledge and the same people. And even the value of the discussions among the experts has been very limited.

I have never understood how string theory manages to stay unitary in the face of the black hole information paradox. Could you say whether this question has been resolved (I have heard differing answers from various string theorists, so I suspect the answer is "no") and if so, how. I know Hawking paid off on his bet, but I'm not satisfied.

Dear Prof Shor, one must be careful what the actual question is.

If the question is whether we have a full mathematical framework that allows us to quantify the problems with Hawking's "proof" of a violation of unitarity in his original language, the answer is No. We know that the violations have something to do with non-locality in the presence of horizons - that almost certainly exists since the argument that it stays exact simply breaks down - but no fully convincing account what happens is known (how much nonlocal it is, how far and quickly one transmits information?), at least it is not known to most people.

The qualitative picture is clear. In the presence of black holes, matter can "tunnel" out of the black hole and the information can do the same. It is known that the exponentially small rate of such tunneling is sufficient to get the information out in time. It's because the difference between mixed states and pure states can be exponentially small, in a proper counting.

But if you ask whether we know that Hawking evaporation is unitary in quantum gravity, according to string theory, the answer is a resounding Yes. There are many clean setups where it can be answered unambiguously - like in AdS5 x S5 or 11-dimensional flat spacetime. One can show that these spacetimes are equivalent to non-gravitational and manifestly unitary theories (the boundary CFT or Matrix theory) with a Hermitean Hamiltonian. We can't yet reformulate these insights in Hawking's original language because the concepts don't match his assumptions. For example, we no longer use a Fock space to describe excitations around a black hole. The black hole microstates themselves differ from a Fock space.

The Preskill-Hawking bet was about the latter, Yes/No question whether the information is actually preserved (not whether Prof Shor would understand the reasons), and the answer is - as both Preskill and Hawking (and your humble correspondent) know today: Yes, the information is preserved.

Surely AdS/CFT was the main reason why Hawking finally agreed that it is preserved. That's why it's completely correct that Hawking has surrendered in the bet regardless whether his own semi-new explanation why the answer is Yes is comprehensible to others or not. His new story is an auxiliary tool that helps him personally to see that loopholes in his previous arguments exist: the main message, that "temporary horizons" shouldn't be treated as God-given mantinels (because the spacetime is asymptotically Minkowski anyway), is surely correct.

More generally, it is clear today that the postulates of quantum mechanics are robust and will survive while the assumptions about exact locality and exact causality don't hold in the presence of horizons where tunneling can occur.

Are we supposed to understand that, somehow, spacetime CONSISTS of strings?

Yes, spacetime somehow consists of strings. Click the link "Why are there gravitons in string theory" above (or here). More precisely, we start with strings as objects propagating on a pre-existing geometry. But without strings, the geometry is rigid. Without strings, there are no additional degrees of freedom that would allow you to change the background you started with.

That changes once you add strings. You find out that if you pump a condensate of strings in a certain vibration mode to the pre-existing spacetime, the physical effect on all other strings and all other objects will be exactly indistinguishable from a deformation of the original geometry. Deformations of spacetimes are equivalent to propagating strings in a certain state. The strings can be poured to space to change the metric tensor. If you wish, you may add or subtract strings from spacetime to get all the way to a vanishing metric tensor.

Also, the consistency criteria for strings to propagate on a background imply that the background must solve Einstein's equations (with all the right corrections). This fact can be proved and also shown to be equivalent to the statements in the previous paragraph about the way how condensates of strings interact.

More questions and answers about this topic can be found in the fast comments.

The punch line is that spacetime itself is created of strings. This conclusion has a very specific content in background-independent string field theory. In this approach to string theory, you may formulate the basic equations of everything roughly as A*A=0 where A is the string field (measures how many strings there are for different shapes at different places, roughly speaking) and * is an operation generalizing the matrix product that knows about the way how strings split and join.

You will find out that there is a classical solution of this equation, A=Q, for every allowed background geometry (Q is the BRST operator that depends on this geometry, roughly speaking, and the equation is equivalent to its nilpotency). You may thus write a general form of A as A=Q+a where a is a small quantum deviation. The equations for a, namely Q*a+a*Q+a*a=0, will describe strings propagating on the background encoded in Q. Note that I could have omitted Q*Q=0. The last, bilinear term in a is the interaction term while the previous terms imply that the state of a string is BRST-closed - the strings satisfy the right equations on the background encoded in Q.

But don't forget that Q itself was made out of strings. There are no other long-distance objects or low-frequency fields except that those you can build out of strings, and strings are enough to reproduce all known types of matter and interactions including gravity in a coherent quantum-mechanical framework. This statement is derived for perturbative string theory that is valid when the coupling is weak. The only change one must make at general couplings is to replace strings with M, whatever it is. But once again, all objects, fields, forces, and dynamics are always made out of the same stuff in string/M-theory, stuff whose properties are completely determined by the mathematically consistent rules.

And that's the memo.

Bonus: The problem for string theorists is that [your humble correspondent] represents all too accurately their views, so they can’t justify censoring him.

I completely agree. A recent problem for string theorists and probably many other scientists is that they are being intimidated by ignorant, dishonest, stupid, bitter, disgusting far left-wing activists such as Peter Woit who wrote the sentence above. They are being threatened and are afraid to say what results science actually leads to: they are effectively being censored and the nasty censors are not even ashamed to admit it. Instead, they are pressured to say things that the human bottom wants them to say. This situation is no science, it is no freedom, and I assure everyone that if this tendency is not fought with, it will become worse and influence science in ever greater number of fields ever more lethally.

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The number 5 refers to the number of fermions that need to be added to the endpoints of strings... But because the 5 fermionic variables are so dummy, the precise arrangement ... is mostly an unphysical artifact of this trick.

Still, I find intriguing that Nature uses 5 fermions at the ends of the QCD string.

Dear Alejandro, thanks. The number 5 is one-half of the number of spacetime dimensions and however mysterious it may sound, it is actually no coincidence. Note that in the bosonic string theory case, the corresponding number instead of 32=2^5 is 8192=2^13 and again, 13 is one-half of 26, the spacetime dimension.

When you calculate the dilaton tadpoles, this relation between the preferred gauge group and the spacetime dimension simply arises. If you make T-duality and work with the middle-dimensional orientifold plans and D-branes, namely D12-branes, O12-planes and/or D4-branes, O4-planes in the superstring case, you will see that one image of the orientifold carries exactly the same tadpole as one image of a half-D-brane. There are 8192 or 32 images of the orientifold in that case. And therefore, the number of half-D-branes must be 8192 or 32, too.

Hello doug, I doubt Lubos would wish to elaborate on sequences, but let me to tell what is known about the two objects you mention.

On 10 and 26, it happens that they are really (9,1) and (25,1). The relevant information is the difference of dimensions, 9-1 and 25-1, and they happen to coincide modulo 8. The roots of this phenomena have mathematical justifications in terms of Clifford Algebras, which in turn is related (separately?) to Bott Periodicity and to selfdual unimodular lattices (The books of Conway come handy here, probably). Looking only to Bott periocicity, one could ask if there is a "missed string theory" in D=18 or even if there is actually a series. Looking to lattices, 10 and 26 are more singular than the rest.

E8 is one of the exceptional groups, and it is not a series but only three: E6, E7, E8. But some authors (Baez tells about this in some posts) like to speak also of "E5, E4", to incorporate SO(10) and SU(3)xSU(2). Mathematicians become furious about this (at least I have a not very good experience with Wodzicki) but GUT-experts are fond of this series. Also, there are some articles beyond "e9, e10,...".

About Fibonacci, well, it is true that the SM particion of flavours in 2 and 3. (u,c and d,s,u) invites to think in a recursive construction, but I do not see how it could proceed. Asking not for recursivity but for self-reference, I noticed that solving for the number of flavours (note Lumos speak of colours) of an "string" (a pair of labels, really) with N generations of Standard Model quarks you get that 2N must be in the series of the even hexagonal numbers (thus 6, 28, 66,...). Ideally one could label as 2+3 (up+down) the five fermions of the Type I superstring and see how the labels combine along the calculations; I can not tell if it can lead somewhere.

(PS: lumo, you surely have noticed I am not using my nick "leucipo". No deep meaning here, only that Blogger is now integrated with Google and it does some auto-autentify trick)